I think that you can't really compare the two. As you have it, Compassionate Christianity is a moral system, while survival of the fittest isn't a system at all, but rather a mechanism. So, it would be better to put something liek Christianity up against some other system and test it using survival of the fittest.
e.g.: Compassionate Christianity vs. Mahayana Buddism.
Or you could look within religions, and go for "Compassionate Christianity" versus "Medieval, Crusading Christianity." With this one, you can see that the compassionate christianity has had a stronger "run" over the last 2k years than the medieval "Ahh, aren't I a nice christian... hey, let's go crusading and wipe out some towns, rape the women and sing soem songs!" Yes, I am generalizing quite a bit, but you get my drift here.
On a sort of related note, my quick and dirty interpretation of the Christianity that Jesus advocated was of the compassionate, pacifist, turn the other cheek variety. Sure you can chase gamblers off of the steps of your temples, but you can't kill them. Even the two quotes often seen in signature lines here where Jesus seems to tell people to buy swords I believe are sarchasm, or something of that sort. The trouble with that form of Christianity is that it does leave you physically vulnerable in the real word we live in. If someone breaks into your house, you aren't supposed to blast them with a shotgun, but instead show them where the valuables are. Spiritually that "system" may work well, but given that we are physical beings that live in this material world, I think that very very very few of us can live like that if we really decided to. What is mor ein line with Jesus' message: having our 2 cars, a computer, $600 custom bowies, etc, or selling all of that stuff and giving the money to the poor so they can eat and get medicine? Trouble is, we want things like computers and cars. And we also think that there is nothing wrong with having a car or using a computer (for the most part.) And that's where you get the "revised" forms of christianity that we have today. I'm sure it took only a few weeks before some of the early christians started to think this way.
Anyways, sorry for the tangent! By the way, though, this should probably be in the community section somewhere. Unless you are asking what kind of hatchet Jesus would use to prepare some wood for Joseph to build a chair out of, in which case I'd say he'd buy a Gransfors Bruks Wildlife Hatchet. They sure look nice!